abstracts-koosis

Han, Hardt, and Lin in the 1990s proved that the size of the singular set of a harmonic function is bounded in terms of its doubling index, which is closely related to the frequency function. In 2017, Naber and Valtorta showed that if a harmonic function in the unit ball in R^d has frequency bounded by N then the (d−2)-dimensional Hausdorff measure of its critical set in the half ball is bounded by exp(CN^2). It is conjectured that the optimal bound should instead be polynomial in N. This is known in dimension two, but the higher-dimensional case remains surprisingly subtle. In this talk, we will review the initial ideas of Han, Hardt, and Lin, and the approach of Naber and Valtorta, we will then sketch some ideas which allow to improve the estimate in dimension three to a bound that is much closer to polynomial. The talk is based on ongoing joint work with Ben Foster and Josef Greilhuber.

Let $X$ be a separable metric space, $\mu$ a Borel measure on $X$, and $B$ some family of open sets of finite positive measure. Define the maximal function of a non-negative integrable function $f$ by
$$
Mf(x)=\sup_\mu(R)^\int_R f\,d\mu\,.
$$
The weak type $L^1$ bound for $M$, i.e., the inequality
$$
\mu(\)\le Ct^\int_X f\,d\mu\,,
$$
in various settings is usually derived from some covering selection property, the most general version of which seems to be as follows:There exist constants $c,C\in(0,+\infty)$ such that for every finite family $B_0\subset B$, there is a subfamily $B_1\subset B_0$ satisfying $\mu(\cup_R)\ge c\mu(\cup_R)$ and $\sum_\chi_R\le C$ $\mu$-almost everywhere.

We shall show that there cannot be any other reason for a weak type $L^1$ bound of the above type, namely, that if the weak type bound holds for the maximal function $M$ associated with the family $B$, then $B$ necessarily has this covering selection property. Time permitting, we'll discuss analogues of this theorem for the similar bounds for $M$ involving $L\log L$ and other Orlich type expressions on the right-hand side.

This is a joint work with Paul Hagelstein and Blanca Radillo-Murguia.

I will talk about the classical gap and type problems in Fourier analysis which focus on functions and measures with gaps in the support of their Fourier transform. These historic problems, studied by many famous mathematicians including Paul Koosis, can be viewed in the context of the uncertainty principle in harmonic analysis. They have multiple connections and applications in various areas of analysis and spectral theory. The second part of the title refers to recently found connections between the type problem and the theory of holographic horizons in high-energy physics.

Directional maximal operators in the plane are governed by the
geometric structure of the underlying set of directions. A central problem
is to characterize those slope sets for which the associated operator is
bounded on nontrivial Lebesgue spaces. Classical results indicate that
finite-order lacunarity plays a decisive role, but earlier formulations
left a gap between combinatorial definitions and the geometric mechanisms
underlying Kakeya-type phenomena.In this talk, I will describe recent joint work with Ed Kroc and Juyoung
Lee that develops a refined notion of admissible finite-order lacunarity.
The key idea is to encode direction sets by rooted, labelled trees and to
interpret lacunarity through structural invariants such as splitting
number. This framework provides a unified way to relate three complementary
perspectives: combinatorial sparsity of the slope set, geometric incidence
properties of associated rectangle families, and analytic boundedness of
directional maximal operators.A central ingredient is a pruning procedure applied to the slope tree,
which produces a subtree of slopes exhibiting controlled Euclidean
separation. This enables a probabilistic construction of Kakeya-type
configurations adapted to the pruned geometry. As a consequence, we obtain
a dichotomy: admissible finite-order lacunary sets give rise to bounded
operators on all nontrivial Lebesgue spaces, while sublacunary sets
generate Kakeya-type phenomena and lead to unboundedness on all such
spaces.

The talk will emphasize the structural ideas behind this approach, as well
as some ongoing directions and open questions.

I shall discuss the following conjecture due to Jean Esterle. For each closed subset $E$ of the unit circle of Lebesgue measure zero, there exists a sequence $u_n\to\infty$ with the following property: if $T$ is any Hilbert-space contraction whose spectrum is contained in $E$, then either $T$ is unitary or $\limsup_\|T^\|/u_n=\infty$. This conjecture is intimately related to the properties of inner functions.